Find "three times a number" first. Then take 7 away from it.
Step-by-step solution
Break the sentence into chunks:
"three times a number" → 3n.
"seven less than" that amount → start with 3n and subtract 7: 3n - 7.
"is 20" → = 20.
The equation is 3n - 7 = 20.
Check: solving gives 3n = 27, n = 9. Read the sentence with 9: three times 9 is 27, and seven less than 27 is 20 ✓
Tip: "Seven less than X" means X - 7. The order in the math is the reverse of the order in the words.
Question 2 of 36
A taxi company charges $3.50 to start a ride plus $2.25 per mile. Which equation gives the total cost C, in dollars, of a ride of m miles?
Hint
Which amount is paid only once? Which amount is paid again for every mile?
Step-by-step solution
The $3.50 is paid once, at the start → it is the constant.
The $2.25 is paid for every mile → it multiplies m: 2.25m.
Add them: C = 2.25m + 3.50 (the same as C = 3.50 + 2.25m).
The equation is C = 3.50 + 2.25m.
Check: With a number, a 4-mile ride should cost 3.50 + 4(2.25) = 3.50 + 9 = \$12.50. The equation gives 3.50 + 2.25(4) = 12.50 ✓. The swapped equation C = 3.50m + 2.25 gives $16.25, which is wrong.
Tip: The per-unit rate always multiplies the variable; the one-time fee stands alone.
Question 3 of 36
Fill in
A new candle is 15 inches tall. When lit, it burns down at a steady 0.5 inch per hour. The height y of the candle, in inches, after x hours is given by an equation of the form y = mx + 15.
Complete the equation: y =x + 15
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Is the height going up or down as time passes? That decides the sign of the number in front of x.
Step-by-step solution
The starting height is 15 inches. That is the constant, already in the equation.
Every hour the candle gets 0.5 inch shorter, so the height changes by -0.5 per hour.
The coefficient of x is the rate of change: m = -0.5.
The equation is y = -0.5x + 15, so the blank is -0.5.
Check: after 2 hours the candle should be 15 - 2(0.5) = 14 inches. The equation gives -0.5(2) + 15 = -1 + 15 = 14 ✓. After 30 hours it is -15 + 15 = 0 (burned out), which makes sense.
Tip: Anything that drains, burns, melts, or is spent has a negative rate. Using +0.5 would make the candle grow taller.
Question 4 of 36
A clue in a puzzle book reads: "Five more than one-fourth of a mystery number k is 12." Which equation matches this clue?
Hint
Build it in pieces: first write "one-fourth of a number," then decide what "five more than" does to it.
Step-by-step solution
"A number" is k.
"One-fourth of a number" means \tfrac{1}{4}times the number: \tfrac{k}{4}.
"Five more than" that amount means add 5 to it: \tfrac{k}{4} + 5.
"is 12" becomes "= 12".
The equation is \dfrac{k}{4} + 5 = 12.
Check: solve it. Subtract 5: \tfrac{k}{4} = 7. Multiply by 4: k = 28. One-fourth of 28 is 7, and five more than 7 is 12 ✓
Tip: "Of" next to a fraction means multiply. Only put parentheses around a group when the sentence says to do something to the whole group (for example, "one-fourth of the sum of a number and five").
Question 5 of 36
A plumber charges an $85 service-call fee plus $60 per hour of work. Which equation gives the total charge C, in dollars, for a job that takes h hours?
Hint
Which amount is paid once, and which amount is paid again for every hour?
Step-by-step solution
The $85 fee is paid once, no matter how long the job takes. It is the constant.
The $60 is paid for each hour, so it multiplies h: 60h.
Total charge = one-time fee + hourly part:
C = 85 + 60h
The equation is C = 85 + 60h.
Check: a 2-hour job costs 85 + 60(2) = 85 + 120 = \$205. That makes sense: $85 to show up, then $60 for each of the 2 hours ✓
Tip: The word per ("per hour," "per mile," "each") marks the rate. The rate always sits next to the variable. A one-time amount is added on its own.
Question 6 of 36
Fill in
A water tank holds 1,200 gallons. A pump drains it at a steady 35 gallons per minute. Write an equation for the volume V of water left in the tank, in gallons, after t minutes.
V =t\ +
Type your answer in the box. Fractions like 3/4 are fine.
Hint
What is the amount at t = 0? Is the volume going up or going down each minute?
Step-by-step solution
Starting value: at t = 0 the tank is full, so V = 1{,}200. This is the constant term.
Rate: every minute the volume drops by 35 gallons. A decrease is a negative rate, so the coefficient of t is -35.
Put it together in the form V = mt + b:
V = -35t + 1{,}200
First blank: -35. Second blank: 1{,}200.
Check: after 10 minutes, V = -35(10) + 1{,}200 = -350 + 1{,}200 = 850 gallons. Draining 35 gallons a minute for 10 minutes removes 350 gallons, and 1{,}200 - 350 = 850 ✓
Tip: Ask "Is the amount getting bigger or smaller?" If it is shrinking, the rate must be negative. V = 1{,}200 - 35t is the same equation written in a different order.
Question 7 of 36
Jamal bought 4 shirts that each cost the same amount and one belt for $12. He spent $96 in all. Which equation can be used to find p, the price of one shirt?
Hint
How much did the 4 shirts cost together, written with p? Then add what else he bought.
Step-by-step solution
One shirt costs p dollars, so 4 shirts cost 4p dollars.
The belt is bought once: $12.
Shirts + belt = total spent:
4p + 12 = 96
The equation is 4p + 12 = 96.
Check: solve it. 4p = 96 - 12 = 84, so p = 21. Four shirts at $21 cost $84, plus the $12 belt is $96 ✓
Tip: Only the item bought several times gets the multiplier. Something bought once (the belt) is added on its own, outside any parentheses.
Question 8 of 36
The perimeter of a rectangle is P = 2l + 2w. Which equation is solved for the width w?
Hint
Treat P and l like numbers. Undo the +\,2l first, then undo the multiplication by 2.
Step-by-step solution
Start with P = 2l + 2w.
Subtract 2l from both sides: P - 2l = 2w.
Divide both whole sides by 2: \dfrac{P - 2l}{2} = w.
The answer is w = \dfrac{P - 2l}{2}.
Check: With numbers, a 5-by-3 rectangle has P = 2(5) + 2(3) = 16. Then w = \dfrac{16 - 2(5)}{2} = \dfrac{6}{2} = 3 ✓
Tip: When you divide, divide the entire side, not just one term. \dfrac{P - 2l}{2} can also be written \dfrac{P}{2} - l.
Question 9 of 36
Drop-down
A school spends exactly $50 on supplies. It buys x notebooks that cost $1.25 each and y pens that cost $0.80 each.
Select the equation that models this situation:
Choose an answer from each menu.
Hint
Each price must be multiplied by the number of that item. The two costs add up to the total.
Step-by-step solution
Cost of the notebooks: (price per notebook) × (number of notebooks) = 1.25x.
Cost of the pens: (price per pen) × (number of pens) = 0.80y.
The two costs together equal the total spent: 1.25x + 0.80y = 50.
The equation is 1.25x + 0.80y = 50.
Check: With numbers, 24 notebooks and 25 pens cost 1.25(24) + 0.80(25) = 30 + 20 = \$50, and the equation gives 50 = 50 ✓. In the swapped equation, the same purchase gives 1.25(25) + 0.80(24) = 31.25 + 19.20 = 50.45 \ne 50.
Tip: Keep each price with its own variable. Write a tiny "label" under each term (notebooks, pens) to be sure.
Question 10 of 36
The volume of a cone is V = \tfrac{1}{3}\pi r^2 h. Which equation is solved for the height h?
Hint
First undo the \tfrac{1}{3} by multiplying both sides by 3. Then undo the multiplication by \pi r^2.
Step-by-step solution
Start with V = \tfrac{1}{3}\pi r^2 h.
Multiply both sides by 3 to undo the \tfrac{1}{3}: 3V = \pi r^2 h.
Divide both sides by \pi r^2: \dfrac{3V}{\pi r^2} = h.
The answer is h = \dfrac{3V}{\pi r^2}.
Check: With numbers, a cone with r = 3 and h = 4 has V = \tfrac{1}{3}\pi(9)(4) = 12\pi. Then \dfrac{3(12\pi)}{\pi(9)} = \dfrac{36\pi}{9\pi} = 4 ✓
Tip: To undo multiplying by \tfrac{1}{3}, multiply by 3 — don't divide by 3. That slip gives \dfrac{V}{3\pi r^2}, which is 9 times too small.
Question 11 of 36
Select all
Maria is 4 years older than twice her son's age. Maria is 38 years old. Let s be her son's age.
Select all the equations that could be used to find s.
Select all that apply.
Hint
First write Maria's age in terms of s. Then look for equations that are rearrangements of the same relationship.
Step-by-step solution
"Twice her son's age" is 2s. "4 years older than" that is 2s + 4.
Maria is 38: 2s + 4 = 38. Solving: 2s = 34, so s = 17.
Now test each choice — a correct choice is the same relationship rearranged, and it must give s = 17:
A.2s + 4 = 38 ✓ (the original equation).
B.2(s + 4) = 38 ✗ — this means "twice (son's age + 4)", giving s = 15.
C.38 - 2s = 4 ✓ — subtract 2s from both sides of A; "Maria's age minus twice the son's age is 4".
D.2s - 4 = 38 ✗ — this makes Maria 4 years younger than twice his age, giving s = 21.
E.s = \dfrac{38 - 4}{2} ✓ — this is A already solved for s: \tfrac{34}{2} = 17.
The correct choices are A, C, and E.
Check: son is 17; twice 17 is 34; 4 more is 38 ✓
Tip: One relationship can be written many ways. Rearranging a correct equation (moving terms, solving for s) gives another correct equation.
Question 12 of 36
Fill in
The simple interest formula is I = prt. Devon's savings account earned $270 in simple interest over 3 years at a rate of 4.5% per year. How much money did Devon put in the account (the principal p)?
$
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Solve I = prt for p, or substitute the numbers first. Write 4.5% as a decimal.
Step-by-step solution
Solve the formula for p: divide both sides by rt. p = \dfrac{I}{rt}.
Write the rate as a decimal: 4.5\% = 0.045.
Substitute: p = \dfrac{270}{0.045 \times 3} = \dfrac{270}{0.135}.
Tip: Always change a percent to a decimal before using I = prt. Using 4.5 instead of 0.045 gives $20, which is far too small for a principal that earns $270.
Question 13 of 36
A car rental company charges $40 per day plus $0.25 per mile. Lena rented a car for 3 days, and her bill was $172.50. Which equation can be used to find m, the number of miles she drove?
Hint
The $40 is charged for each day. The $0.25 is charged for each mile.
Step-by-step solution
Daily charge: $40 for each of 3 days = 3(40).
Mileage charge: $0.25 for each of m miles = 0.25m.
Total: 3(40) + 0.25m = 172.50.
The equation is 3(40) + 0.25m = 172.50.
Check: By solving, 120 + 0.25m = 172.50, so 0.25m = 52.50 and m = 210 miles. Then 3(40) + 0.25(210) = 120 + 52.50 = \$172.50 ✓
Tip: Ask "what is each rate multiplied by?" — days for the daily rate, miles for the mileage rate. Only the rate that matches the unknown gets the variable.
Question 14 of 36
At a bake sale, muffins sell for $2.50 each and cookies sell for $1.75 each. The sale brought in exactly $210. Let m be the number of muffins sold and c the number of cookies sold. Which equation models this situation?
Hint
Money from muffins + money from cookies = total money. How do you find the money from m muffins?
Step-by-step solution
Money from muffins: each muffin brings $2.50, so m muffins bring 2.50m dollars.
Money from cookies: each cookie brings $1.75, so c cookies bring 1.75c dollars.
The two amounts together make the total of $210:
2.50m + 1.75c = 210
The equation is 2.50m + 1.75c = 210.
Check: test a possible pair. If 56 muffins and 40 cookies were sold: 2.50(56) + 1.75(40) = 140 + 70 = 210 ✓ — so the equation describes real combinations of sales.
Tip: In a "total value" equation, each price multiplies the number of that item. The right side is a dollar amount, so every term on the left must also be dollars.
Question 15 of 36
The formula F = \dfrac{9}{5}C + 32 converts a temperature in degrees Celsius (C) to degrees Fahrenheit (F). Which equation is solved for C?
Hint
To get C alone, undo the operations in reverse order: the "+ 32" was done last, so undo it first.
Step-by-step solution
Undo the operations on C in reverse order.
Start: F = \tfrac{9}{5}C + 32.
The last thing done to C was "add 32," so subtract 32 from both sides: F - 32 = \tfrac{9}{5}C.
C is multiplied by \tfrac{9}{5}. Multiply both sides by the reciprocal, \tfrac{5}{9}:
\tfrac{5}{9}(F - 32) = C
The equation is C = \dfrac{5}{9}(F - 32).
Check: water boils at 212^\circF. \tfrac{5}{9}(212 - 32) = \tfrac{5}{9}(180) = 100^\circC ✓. And \tfrac{9}{5}(100) + 32 = 180 + 32 = 212 ✓
Tip: The parentheses matter. The \tfrac{5}{9} must multiply the wholeF - 32, because you subtracted 32 before you undid the fraction.
Question 16 of 36
A lake shop rents kayaks. The table shows the total rental cost y, in dollars, for x hours.
Hours, x
0
1
2
3
Cost, y ($)
15
23
31
39
Which equation represents the relationship in the table?
Hint
How much does the cost go up for each extra hour? What is the cost when x = 0?
Step-by-step solution
Rate: each time x goes up by 1 hour, y goes up by $8: 23 - 15 = 8, 31 - 23 = 8, 39 - 31 = 8. So m = 8.
Starting value: when x = 0, y = 15. So b = 15 (a $15 fee before any time is used).
Write y = mx + b:
y = 8x + 15
The equation is y = 8x + 15.
Check: test the last row: 8(3) + 15 = 24 + 15 = 39 ✓. Test x = 2: 16 + 15 = 31 ✓
Tip: In a table, the rate is (change in y) ÷ (change in x), and the starting value is the y in the row where x = 0. Always test your equation on a row you did not use.
Question 17 of 36
The formula A = P + Prt gives the amount A in an account after simple interest. Rosa tried to solve it for r. Her work is below.
Step 1:A - P = Prt
Step 2:A - P - P = rt
Step 3:r = \dfrac{A - 2P}{t}
In which step did Rosa make her first mistake?
Hint
Check each step: is the same operation done to both sides, and is it the right inverse operation?
Step-by-step solution
Check each step.
Step 1: subtracting P from both sides of A = P + Prt gives A - P = Prt. Correct.
Step 2: on the right, P is multiplied by rt. To remove it you must divide both sides by P, not subtract P. Rosa subtracted, so Step 2 is the first mistake.
Correct version: \dfrac{A - P}{P} = rt, then divide by t:
r = \dfrac{A - P}{Pt}
Rosa's first mistake is in Step 2. (Step 3 follows from Step 2, but it is wrong only because Step 2 was wrong.)
Check: with P = 1{,}000, r = 0.05, t = 2: A = 1{,}000 + 1{,}000(0.05)(2) = 1{,}100. The correct formula gives r = \dfrac{100}{2{,}000} = 0.05 ✓. Rosa's gives \dfrac{1{,}100 - 2{,}000}{2} = -450 ✗
Tip: The inverse of multiplication is division. A letter stuck to other letters by multiplication (Prt) is removed by dividing, never by subtracting.
Question 18 of 36
Select all
The sum of three consecutive even integers is 78. Let n be the smallest of the three integers.
Select all the equations that could be used to find n.
Select all that apply.
Hint
Consecutive even integers are 2 apart. Write the three numbers using n, then look for equations that are the same relationship written another way.
Step-by-step solution
Consecutive even integers go up by 2: the three numbers are n, n + 2, and n + 4.
Their sum is 78: n + (n + 2) + (n + 4) = 78. (A) is correct.
Combine like terms: 3n + 6 = 78. (C) is correct.
Factor out 3: 3n + 6 = 3(n + 2), so 3(n + 2) = 78. (E) is correct. (It says "3 times the middle number is 78.")
(B) uses n + 1 and n + 2, which are consecutive integers, not consecutive even integers.
(D) leaves out the +2 and +4.
Correct: A, C, and E.
Check: solve 3n + 6 = 78: 3n = 72, n = 24. The numbers are 24, 26, 28, all even, and 24 + 26 + 28 = 78 ✓
Tip: For consecutive even or odd integers, add 2 each time. For plain consecutive integers, add 1.
Question 19 of 36
Drop-down
An electrician charges a $55 trip fee for each visit plus $72 per hour of work. Complete the equation for the total charge C, in dollars, for a visit of h hours.
C =h\ +
Choose an answer from each menu.
Hint
The number that multiplies h is paid for every hour. The number added alone is paid once per visit.
Step-by-step solution
Rate (multiplies h): $72 is charged for each hour, so the coefficient of h is 72.
Constant: the $55 trip fee is charged once per visit, so it is added on its own.
C = 72h + 55
First blank: 72. Second blank: 55.
Check: a 3-hour visit costs 72(3) + 55 = 216 + 55 = \$271: three hours of work at $72, plus one trip fee ✓
Tip: Read the words around each number. "Per hour" → coefficient of h. "Fee," "flat," "one-time," "for each visit" (when h counts hours, not visits) → constant.
Question 20 of 36
Fill in
An outdoor concert sells lawn tickets for $30 each and reserved-seat tickets for $55 each. Ticket sales totaled $16,500. Let x be the number of lawn tickets and y the number of reserved-seat tickets sold. Complete the equation that models the ticket sales.
x\ +y =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Each kind of ticket brings in (price) × (number sold) dollars. What do the two amounts add up to?
Step-by-step solution
Lawn tickets: x tickets at $30 each bring in 30x dollars.
Reserved seats: y tickets at $55 each bring in 55y dollars.
Together they make the total sales, $16,500:
30x + 55y = 16{,}500
Blanks: 30, 55, 16500.
Check: test one possible combination, 330 lawn tickets and 120 reserved seats: 30(330) + 55(120) = 9{,}900 + 6{,}600 = 16{,}500 ✓. So the equation is true for real combinations of sales.
Tip: Keep each price glued to its own variable — the problem tells you x is lawn, so 30 goes with x. Commas in large numbers are just for reading: type 16500.
Question 21 of 36
No calculatorFill in
Rewrite the equation 5x + 2y = 14 in slope-intercept form, y = mx + b.
y =x\ +
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Move the x-term to the other side first. Then divide every term by the number in front of y.
Step-by-step solution
Start: 5x + 2y = 14.
Subtract 5x from both sides: 2y = -5x + 14. (The 5x becomes -5x when it changes sides.)
Divide every term by 2:
y = -\dfrac{5}{2}x + 7
m = -2.5 (or -\tfrac{5}{2}) and b = 7.
Check: pick x = 2. Then y = -5 + 7 = 2. In the original: 5(2) + 2(2) = 10 + 4 = 14 ✓. Pick x = 0: y = 7, and 2(7) = 14 ✓
Tip: When you divide by the coefficient of y, divide the x-term and the constant. Forgetting the constant leaves b = 14 instead of 7.
Question 22 of 36
Which equation is equivalent to 3x - 4y = 12 and is solved for y?
Hint
Move the x term to the right side first. Then divide every term by the coefficient of y, including its negative sign.
Step-by-step solution
Subtract 3x from both sides: -4y = -3x + 12.
Divide every term by -4: y = \dfrac{-3x}{-4} + \dfrac{12}{-4}.
Simplify the signs: \dfrac{-3}{-4} = \dfrac{3}{4} and \dfrac{12}{-4} = -3.
The answer is y = \dfrac{3}{4}x - 3.
Check: With a point, if x = 4, then y = \tfrac{3}{4}(4) - 3 = 0. In the original: 3(4) - 4(0) = 12 ✓. If x = 0, y = -3: 3(0) - 4(-3) = 12 ✓
Tip: Dividing by a negative changes the sign of every term. Each wrong choice here comes from changing only one sign, or none.
Question 23 of 36
Drag and drop
Drag the equation that models each situation into its box. In each equation, x is the number of weeks or items and y is the total. One equation is not used.
A plant is 4 inches tall and grows 2 inches per week.
A savings account has $400, and $20 is withdrawn each week.
T-shirts cost $20 each, plus a $4 shipping charge per order.
Tap a tile, then tap a box to place it.
Hint
For each situation, find the starting (one-time) amount and the amount that changes per week or per item. Is the change up or down?
Step-by-step solution
For each situation, identify the start value (constant) and the rate (coefficient of x):
Plant: starts at 4 in, grows 2 in per week → rate +2, start 4 → y = 2x + 4.
Savings: starts at $400, $20 taken out each week → rate -20, start 400 → y = 400 - 20x.
T-shirts: $20 per shirt, $4 once per order → rate 20, constant 4 → y = 20x + 4.
The leftover equation, y = 4x + 2, swaps the plant's rate and starting height. It would mean a 2-inch plant growing 4 inches a week, so it is not used.
Answer: plant → y = 2x + 4; savings → y = 400 - 20x; T-shirts → y = 20x + 4.
Check: after 3 weeks the plant is 4 + 3(2) = 10 in, and 2(3) + 4 = 10 ✓; after 3 weeks the account has 400 - 60 = \$340 ✓; 3 shirts cost 60 + 4 = \$64 ✓
Tip: The number that repeats goes with x. The number that happens once is the constant. Decreasing amounts get a minus sign.
Question 24 of 36
A delivery van starts the day with 14 gallons of gas. It uses 1 gallon for every 28 miles it drives. Which equation gives the number of gallons g left in the tank after the van drives m miles?
Hint
How many gallons does the van use for one mile? Is the amount of gas going up or down?
Step-by-step solution
Rate per mile: 1 gallon for every 28 miles means \tfrac{1}{28} gallon per mile. So m miles use \tfrac{m}{28} gallons.
Direction: the gas is being used up, so subtract.
Start: 14 gallons.
g = 14 - \dfrac{m}{28}
The equation is g = 14 - \dfrac{m}{28}.
Check: after 140 miles, g = 14 - \tfrac{140}{28} = 14 - 5 = 9 gallons. 140 miles at 28 miles per gallon uses 5 gallons ✓. The tank is empty when \tfrac{m}{28} = 14, at m = 392 miles ✓
Tip: "1 gallon per 28 miles" and "28 miles per gallon" mean the same thing. The rate you need is per mile (the unit of the variable), which is \tfrac{1}{28}, not 28.
Question 25 of 36
The surface area of a cylinder is SA = 2\pi rh + 2\pi r^2. Which equation is solved for the height h?
Hint
Only one term contains h. Move the other term away first, then divide by everything that multiplies h.
Step-by-step solution
Start: SA = 2\pi rh + 2\pi r^2. Only the term 2\pi rh contains h.
Subtract the other term, 2\pi r^2, from both sides: SA - 2\pi r^2 = 2\pi rh.
h is multiplied by 2\pi r. Divide both whole sides by 2\pi r:
h = \dfrac{SA - 2\pi r^2}{2\pi r}
The equation is h = \dfrac{SA - 2\pi r^2}{2\pi r}.
Check: a can with r = 1 and h = 3 has SA = 2\pi(1)(3) + 2\pi(1)^2 = 6\pi + 2\pi = 8\pi. The formula gives h = \dfrac{8\pi - 2\pi}{2\pi} = \dfrac{6\pi}{2\pi} = 3 ✓
Tip: When you divide, the fraction bar must go under the entire other side. Dividing only SA by 2\pi r (and leaving 2\pi r^2 outside) is the most common error on this kind of question.
Question 26 of 36
Fill in
A cylindrical water tank holds 942 cubic feet of water when full. Its radius is 5 feet. The volume of a cylinder is V = \pi r^2 h. Using \pi \approx 3.14, what is the height of the tank?
h = feet
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Solve V = \pi r^2 h for h first, then substitute. Remember that r^2 means r \times r.
Step-by-step solution
Solve the formula for h: h is multiplied by \pi r^2, so divide both sides by \pi r^2:
Tip: A common slip is to use 2r = 10 instead of r^2 = 25 (that gives 30 ft), or to forget to square at all (that gives 60 ft). On the calculator, type 942 ÷ ( 3.14 × 5 x² ) with parentheses around the whole bottom.
Question 27 of 36
A hotel charges $129 per night for a room, plus a parking fee of $15 per night, plus a one-time cleaning fee of $40. Ming’s total bill was $616. Which equation can be used to find n, the number of nights she stayed?
Hint
Which charges repeat every night, and which charge happens only once?
Step-by-step solution
Charges for each night: room $129 and parking $15, so each night costs 129 + 15 = \$144. For n nights: 144n.
One-time charge: the $40 cleaning fee is added once.
Total bill:
144n + 40 = 616
The equation is 144n + 40 = 616.
Check: solve it. 144n = 576, so n = 4 nights. Room: 4 \times 129 = 516; parking: 4 \times 15 = 60; cleaning: 40. Total 516 + 60 + 40 = \$616 ✓
Tip: Group all the per-night amounts together before multiplying by n. Then add the one-time amounts at the end.
Question 28 of 36
Drag and drop
Drag the equation that models each situation into its box. In each equation, x is the number of months, hours, or visits, and y is the total. One equation is not used.
A gym charges a $25 joining fee and $15 per month.
A tank holds 150 gallons, and 25 gallons are pumped out each hour.
A tutor earns $25 per visit plus a $15 bonus, paid once.
Tap a tile, then tap a box to place it.
Hint
For each situation, find the amount that happens once and the amount that repeats. Does the total go up or down?
Step-by-step solution
For each situation, find the constant (one-time amount) and the rate (amount per month, hour, or visit).
Gym: $25 once, $15 each month → rate 15, constant 25 → y = 15x + 25.
Tank: starts at 150 gallons and loses 25 each hour → rate -25, constant 150 → y = 150 - 25x.
Tutor: $25 each visit, $15 once → rate 25, constant 15 → y = 25x + 15.
The leftover tile, y = 25x + 150, would make the tank gain 25 gallons every hour, so it is not used.
Answer: gym → y = 15x + 25; tank → y = 150 - 25x; tutor → y = 25x + 15.
Check: after 2 months the gym costs 25 + 30 = \$55, and 15(2) + 25 = 55 ✓. After 2 hours the tank has 150 - 50 = 100 gallons ✓. After 2 visits the tutor has earned 50 + 15 = \$65, and 25(2) + 15 = 65 ✓
Tip: The same two numbers (15 and 25) can make two different equations. Decide which one repeats: that one multiplies x.
Question 29 of 36
Fill in
Dana saves the same amount of money every week. The table shows her savings y, in dollars, after x weeks.
Week, x
2
5
8
11
Savings, y ($)
340
475
610
745
Write an equation for her savings in the form y = mx + b.
y =x\ +
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The weeks go up by 3, not by 1. Find the change per one week, then work backwards from week 2 to week 0.
Step-by-step solution
Rate: from week 2 to week 5, savings rise 475 - 340 = 135 dollars in 5 - 2 = 3 weeks. Per week: \tfrac{135}{3} = 45. So m = 45.
Starting value: the table does not show week 0, so work backwards. From week 2 back to week 0 is 2 weeks, so subtract 2 \times 45 = 90: 340 - 90 = 250. So b = 250.
Tip: Two traps here. (1) The x-values jump by 3, so the change of 135 is not the weekly rate. (2) The first row is week 2, not week 0, so 340 is not the starting value. Always find b at x = 0.
Question 30 of 36
The length of a rectangular dog run is 3 feet more than twice its width. The perimeter of the dog run is 54 feet. Which equation can be used to find the width w, in feet?
Hint
Write the length in terms of w first. A rectangle has two lengths and two widths.
Step-by-step solution
"Twice the width" is 2w. "3 feet more than" that is 2w + 3. So the length is \ell = 2w + 3.
Perimeter of a rectangle: P = 2\ell + 2w (two lengths plus two widths).
Substitute \ell = 2w + 3 and P = 54:
2(2w + 3) + 2w = 54
The equation is 2(2w + 3) + 2w = 54.
Check: solve it. 4w + 6 + 2w = 54, so 6w = 48 and w = 8 ft. Length = 2(8) + 3 = 19 ft. Perimeter = 2(19) + 2(8) = 38 + 16 = 54 ft ✓
Tip: Perimeter goes all the way around, so every side counts. Area (length × width) is a different measure — don't mix them up when the problem says "perimeter."
Question 31 of 36
After an oven is turned off, its temperature drops at a steady rate. The table shows the temperature T, in °F, m minutes after it was turned off.
Minutes, m
3
7
10
15
Temperature, T (°F)
314
266
230
170
Which equation models the oven temperature?
Hint
The minutes are not evenly spaced. Divide each change in temperature by the matching change in minutes. Then work back to m = 0.
Step-by-step solution
Rate: from 3 to 7 minutes, T drops 314 - 266 = 48° in 7 - 3 = 4 minutes: \tfrac{-48}{4} = -12° per minute. Confirm with another pair: from 10 to 15 minutes, \tfrac{170 - 230}{5} = \tfrac{-60}{5} = -12 ✓
Starting value: go back 3 minutes from m = 3: 314 + 3 \times 12 = 314 + 36 = 350°F.
Tip: With uneven steps in a table, never use a single jump in y as the rate — always divide by the jump in x. And the first row shown is not always x = 0.
Question 32 of 36
A landscaper charges $45 per hour plus a $70 equipment fee for each job. One bill was $295. Sam wrote and solved an equation to find the number of hours h:
Look at Sam’s equation before looking at his algebra. Which number is paid per hour?
Step-by-step solution
Sam's algebra is fine: from 70h + 45 = 295, subtracting 45 gives 70h = 250, and 250 \div 70 \approx 3.57.
The problem is the equation. The $45 is charged per hour, so it multiplies h. The $70 fee is charged once, so it is the constant. Sam swapped them.
Correct equation: 45h + 70 = 295.
Solve: 45h = 225, so h = 5.
Sam swapped the hourly rate and the one-time fee; the job took 5 hours.
Check:45(5) + 70 = 225 + 70 = 295 ✓. A non-whole answer like 3.57 hours was also a warning sign that the setup might be wrong.
Tip: Before solving, read your equation back in words: "70 dollars each hour plus 45 once" — does that match the story? If not, fix the equation first.
Question 33 of 36
Fill in
The trapezoid below has an area of 84 square centimeters.
The area of a trapezoid is A = \tfrac{1}{2}h(b_1 + b_2). What is the length of the base b_2?
b_2 = cm
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Solve the formula for b_2: undo the \tfrac{1}{2}, then the h, then the b_1. Or substitute the numbers first and solve.
Step-by-step solution
Solve the formula for b_2 first.
Multiply both sides by 2: 2A = h(b_1 + b_2).
Divide both sides by h: \dfrac{2A}{h} = b_1 + b_2.
Tip: Undo the operations in reverse order: the \tfrac{1}{2} and the h multiply the whole sum, so remove them before subtracting b_1. Forgetting to multiply by 2 gives 10.5 - 9 = 1.5, which is far too small for the picture.
Question 34 of 36
Rewrite the line \dfrac{2}{3}x - \dfrac{1}{2}y = 4 in slope-intercept form. Which of the following is the result?
Hint
Move the x-term first. Then undo "times -\tfrac{1}{2}" by multiplying every term by -2.
Step-by-step solution
Start: \tfrac{2}{3}x - \tfrac{1}{2}y = 4.
Subtract \tfrac{2}{3}x from both sides: -\tfrac{1}{2}y = -\tfrac{2}{3}x + 4.
y is multiplied by -\tfrac{1}{2}. Multiply every term by the reciprocal, -2:
y = (-2)\left(-\tfrac{2}{3}x\right) + (-2)(4) = \tfrac{4}{3}x - 8
The equation is y = \dfrac{4}{3}x - 8.
Check: choose x = 6. Then y = 8 - 8 = 0. In the original: \tfrac{2}{3}(6) - \tfrac{1}{2}(0) = 4 ✓. Choose x = 0: y = -8, and -\tfrac{1}{2}(-8) = 4 ✓
Tip: To undo multiplying by a fraction, multiply by its reciprocal (flip it, keep the sign). Multiplying by -2 changes the sign of both terms on the right.
Question 35 of 36
Drag and drop
A community theater sells adult tickets for $12 and child tickets for $7. Let x be the number of adult tickets and y the number of child tickets. Drag the equation that matches each fact into its box. Two equations are not used.
The ticket sales brought in $4,145.
400 tickets were sold in all.
The number of adult tickets was 7 more than twice the number of child tickets.
Tap a tile, then tap a box to place it.
Hint
Ask what each fact counts: dollars or tickets? For the last fact, "twice the number of child tickets" is built first.
Step-by-step solution
$4,145 in sales counts dollars: each adult ticket brings $12 and each child ticket $7, so 12x + 7y = 4{,}145.
400 tickets counts tickets, not dollars: x + y = 400.
Adult tickets were 7 more than twice the child tickets: "twice the child tickets" is 2y; "7 more than" that is 2y + 7; so x = 2y + 7.
Unused tiles: 7x + 12y = 4{,}145 puts the prices on the wrong tickets; x = 7y + 2 swaps "7 more" and "twice."
Answer: $4,145 → 12x + 7y = 4{,}145; 400 tickets → x + y = 400; 7 more than twice → x = 2y + 7.
Check: all three facts fit together: 131 child tickets and 2(131) + 7 = 269 adult tickets make 269 + 131 = 400 tickets and 12(269) + 7(131) = 3{,}228 + 917 = \$4{,}145 ✓. For the last fact, if 10 child tickets were sold, there would be 2(10) + 7 = 27 adult tickets — 7 more than 20 ✓. The tile x = 7y + 2 would give 72, which is "2 more than 7 times."
Tip: In "a more than b times a number," the b multiplies the variable and the "more than" number a is added: x = by + a.
Question 36 of 36
Theo solved the equation ax + 3 = bx - 7 for x (assume a \ne b). His work is below.
Step 1:ax - bx + 3 = -7
Step 2:ax - bx = -10
Step 3:x(a - b) = -10
Step 4:x = \dfrac{-10}{a} - b
In which step did Theo make his first mistake?
Hint
Check each step with simple numbers, for example a = 3 and b = 1.
Step-by-step solution
Check each step.
Step 1: subtract bx from both sides → ax - bx + 3 = -7. Correct.
Step 2: subtract 3 from both sides → ax - bx = -10. Correct.
Step 3: factor x out of ax - bx → x(a - b) = -10. Correct.
Step 4: to undo "times (a - b)" you divide by the whole quantity a - b: x = \dfrac{-10}{a - b}. Theo divided by a only and then subtracted b. This is the first mistake.
Theo's first mistake is in Step 4.
Check: try a = 3, b = 1: the equation is 3x + 3 = x - 7, so 2x = -10 and x = -5. The correct formula gives \tfrac{-10}{3 - 1} = -5 ✓. Theo's gives \tfrac{-10}{3} - 1 \approx -4.33 ✗
Tip: After you factor out the variable, the whole parentheses is one number. Divide by all of it, and write the fraction bar under the whole (a - b).
